Theorem signslema 26978

Description: Computational part of signwlemn . (Contributed by Thierry Arnoux, 29-Sep-2018.)

Hypotheses

Ref	Expression
signslema.1	|- ( ph -> E e. NN0 )
signslema.2	|- ( ph -> F e. NN0 )
signslema.3	|- ( ph -> G e. NN0 )
signslema.4	|- ( ph -> H e. NN0 )
signslema.5	|- ( ph -> ( E < G /\ -. 2 || ( G - E ) ) )
signslema.6	|- ( ph -> ( ( H - G ) - ( F - E ) ) e. { 0 , 2 } )

Assertion

Ref	Expression
signslema	|- ( ph -> ( F < H /\ -. 2 || ( H - F ) ) )

Proof of Theorem signslema

Step	Hyp	Ref	Expression
1		signslema.5	. . . . . 6 |- ( ph -> ( E < G /\ -. 2 || ( G - E ) ) )
2	1	simpld 459	. . . . 5 |- ( ph -> E < G )
3	2	adantr 465	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> E < G )
4		signslema.4	. . . . . . . . . 10 |- ( ph -> H e. NN0 )
5	4	nn0cnd 10653	. . . . . . . . 9 |- ( ph -> H e. CC )
6		signslema.2	. . . . . . . . . 10 |- ( ph -> F e. NN0 )
7	6	nn0cnd 10653	. . . . . . . . 9 |- ( ph -> F e. CC )
8	5, 7	subcld 9734	. . . . . . . 8 |- ( ph -> ( H - F ) e. CC )
9		signslema.3	. . . . . . . . . 10 |- ( ph -> G e. NN0 )
10	9	nn0cnd 10653	. . . . . . . . 9 |- ( ph -> G e. CC )
11		signslema.1	. . . . . . . . . 10 |- ( ph -> E e. NN0 )
12	11	nn0cnd 10653	. . . . . . . . 9 |- ( ph -> E e. CC )
13	10, 12	subcld 9734	. . . . . . . 8 |- ( ph -> ( G - E ) e. CC )
14	8, 13	subeq0ad 9744	. . . . . . 7 |- ( ph -> ( ( ( H - F ) - ( G - E ) ) = 0 <-> ( H - F ) = ( G - E ) ) )
15	14	biimpa 484	. . . . . 6 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( H - F ) = ( G - E ) )
16	15	breq2d 4319	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( 0 < ( H - F ) <-> 0 < ( G - E ) ) )
17	6	nn0red 10652	. . . . . . 7 |- ( ph -> F e. RR )
18	4	nn0red 10652	. . . . . . 7 |- ( ph -> H e. RR )
19	17, 18	posdifd 9941	. . . . . 6 |- ( ph -> ( F < H <-> 0 < ( H - F ) ) )
20	19	adantr 465	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( F < H <-> 0 < ( H - F ) ) )
21	11	nn0red 10652	. . . . . . 7 |- ( ph -> E e. RR )
22	9	nn0red 10652	. . . . . . 7 |- ( ph -> G e. RR )
23	21, 22	posdifd 9941	. . . . . 6 |- ( ph -> ( E < G <-> 0 < ( G - E ) ) )
24	23	adantr 465	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( E < G <-> 0 < ( G - E ) ) )
25	16, 20, 24	3bitr4rd 286	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( E < G <-> F < H ) )
26	3, 25	mpbid 210	. . 3 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> F < H )
27		0re 9401	. . . . . 6 |- 0 e. RR
28	27	a1i 11	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> 0 e. RR )
29	22, 21	resubcld 9791	. . . . . 6 |- ( ph -> ( G - E ) e. RR )
30	29	adantr 465	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( G - E ) e. RR )
31	18, 17	resubcld 9791	. . . . . 6 |- ( ph -> ( H - F ) e. RR )
32	31	adantr 465	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( H - F ) e. RR )
33	2	adantr 465	. . . . . 6 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> E < G )
34	23	adantr 465	. . . . . 6 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( E < G <-> 0 < ( G - E ) ) )
35	33, 34	mpbid 210	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> 0 < ( G - E ) )
36		2pos 10428	. . . . . . . 8 |- 0 < 2
37		breq2 4311	. . . . . . . 8 |- ( ( ( H - F ) - ( G - E ) ) = 2 -> ( 0 < ( ( H - F ) - ( G - E ) ) <-> 0 < 2 ) )
38	36, 37	mpbiri 233	. . . . . . 7 |- ( ( ( H - F ) - ( G - E ) ) = 2 -> 0 < ( ( H - F ) - ( G - E ) ) )
39	38	adantl 466	. . . . . 6 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> 0 < ( ( H - F ) - ( G - E ) ) )
40	29, 31	posdifd 9941	. . . . . . 7 |- ( ph -> ( ( G - E ) < ( H - F ) <-> 0 < ( ( H - F ) - ( G - E ) ) ) )
41	40	biimpar 485	. . . . . 6 |- ( ( ph /\ 0 < ( ( H - F ) - ( G - E ) ) ) -> ( G - E ) < ( H - F ) )
42	39, 41	syldan 470	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( G - E ) < ( H - F ) )
43	28, 30, 32, 35, 42	lttrd 9547	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> 0 < ( H - F ) )
44	19	adantr 465	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( F < H <-> 0 < ( H - F ) ) )
45	43, 44	mpbird 232	. . 3 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> F < H )
46	5, 10, 7, 12	sub4d 9783	. . . . 5 |- ( ph -> ( ( H - G ) - ( F - E ) ) = ( ( H - F ) - ( G - E ) ) )
47		signslema.6	. . . . 5 |- ( ph -> ( ( H - G ) - ( F - E ) ) e. { 0 , 2 } )
48	46, 47	eqeltrrd 2518	. . . 4 |- ( ph -> ( ( H - F ) - ( G - E ) ) e. { 0 , 2 } )
49		ovex 6131	. . . . 5 |- ( ( H - F ) - ( G - E ) ) e. _V
50	49	elpr 3910	. . . 4 |- ( ( ( H - F ) - ( G - E ) ) e. { 0 , 2 } <-> ( ( ( H - F ) - ( G - E ) ) = 0 \/ ( ( H - F ) - ( G - E ) ) = 2 ) )
51	48, 50	sylib 196	. . 3 |- ( ph -> ( ( ( H - F ) - ( G - E ) ) = 0 \/ ( ( H - F ) - ( G - E ) ) = 2 ) )
52	26, 45, 51	mpjaodan 784	. 2 |- ( ph -> F < H )
53	1	simprd 463	. . . . 5 |- ( ph -> -. 2 || ( G - E ) )
54	53	adantr 465	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> -. 2 || ( G - E ) )
55	15	breq2d 4319	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( 2 || ( H - F ) <-> 2 || ( G - E ) ) )
56	54, 55	mtbird 301	. . 3 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> -. 2 || ( H - F ) )
57		2z 10693	. . . . . . 7 |- 2 e. ZZ
58	9	nn0zd 10760	. . . . . . . 8 |- ( ph -> G e. ZZ )
59	11	nn0zd 10760	. . . . . . . 8 |- ( ph -> E e. ZZ )
60	58, 59	zsubcld 10767	. . . . . . 7 |- ( ph -> ( G - E ) e. ZZ )
61		dvdsaddr 13587	. . . . . . 7 |- ( ( 2 e. ZZ /\ ( G - E ) e. ZZ ) -> ( 2 || ( G - E ) <-> 2 || ( ( G - E ) + 2 ) ) )
62	57, 60, 61	sylancr 663	. . . . . 6 |- ( ph -> ( 2 || ( G - E ) <-> 2 || ( ( G - E ) + 2 ) ) )
63	53, 62	mtbid 300	. . . . 5 |- ( ph -> -. 2 || ( ( G - E ) + 2 ) )
64	63	adantr 465	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> -. 2 || ( ( G - E ) + 2 ) )
65		2cnd 10409	. . . . . . . 8 |- ( ph -> 2 e. CC )
66	8, 13, 65	subaddd 9752	. . . . . . 7 |- ( ph -> ( ( ( H - F ) - ( G - E ) ) = 2 <-> ( ( G - E ) + 2 ) = ( H - F ) ) )
67	66	biimpa 484	. . . . . 6 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( ( G - E ) + 2 ) = ( H - F ) )
68	67	breq2d 4319	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( 2 || ( ( G - E ) + 2 ) <-> 2 || ( H - F ) ) )
69	68	notbid 294	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( -. 2 || ( ( G - E ) + 2 ) <-> -. 2 || ( H - F ) ) )
70	64, 69	mpbid 210	. . 3 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> -. 2 || ( H - F ) )
71	56, 70, 51	mpjaodan 784	. 2 |- ( ph -> -. 2 || ( H - F ) )
72	52, 71	jca 532	1 |- ( ph -> ( F < H /\ -. 2 || ( H - F ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1369   e. wcel 1756   {cpr 3894   class class class wbr 4307  (class class class)co 6106   RRcr 9296   0cc0 9297   + caddc 9300   < clt 9433   - cmin 9610   2c2 10386   NN0cn0 10594   ZZcz 10661   || cdivides 13550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-recs 6847  df-rdg 6881  df-er 7116  df-en 7326  df-dom 7327  df-sdom 7328  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-nn 10338  df-2 10395  df-n0 10595  df-z 10662  df-dvds 13551

This theorem is referenced by: (None)